This article provides a comprehensive overview of recent developments in continuous-time quantum Monte Carlo (CT-QMC) methods for solving quantum impurity models. The authors present detailed derivations and descriptions of the algorithms, enabling others to implement them. They discuss the strengths and weaknesses of the methods, their range of applicability, and summarize the problems where these methods have been successfully applied. The review also outlines future prospects for their use. Quantum impurity models describe an atom or molecule embedded in a host material, and are fundamental in nanoscience and correlated electron materials. These models are used to study quantum dots, molecular conductors, and the Kondo effect. The CT-QMC methods discussed here are particularly effective for a wide range of physically realistic models, providing access to both high and low energy scales. The article reviews several CT-QMC methods, including diagrammatic Monte Carlo, interaction expansion (CT-INT), auxiliary field (CT-AUX), hybridization expansion (CT-HYB), and infinite-U (CT-J) methods. Each method is described in detail, with their strengths and limitations discussed. The methods are particularly useful for solving problems involving many-body interactions, such as the Anderson impurity model and the Kondo lattice model. The review also discusses the challenges of simulating quantum impurity models, including the sign problem, which arises due to the complex nature of fermionic interactions. The CT-QMC methods are particularly effective in overcoming these challenges, allowing for accurate simulations of a wide range of physical systems. The article highlights the importance of CT-QMC methods in modern condensed matter physics, particularly in the study of strongly correlated electron systems. These methods have enabled the accurate simulation of complex systems, including those with multiple orbitals and interactions, and have been applied to a variety of problems in nanoscience and nonequilibrium physics. The review concludes with a discussion of the future prospects for CT-QMC methods, emphasizing their potential for further advancements in the study of quantum impurity models.This article provides a comprehensive overview of recent developments in continuous-time quantum Monte Carlo (CT-QMC) methods for solving quantum impurity models. The authors present detailed derivations and descriptions of the algorithms, enabling others to implement them. They discuss the strengths and weaknesses of the methods, their range of applicability, and summarize the problems where these methods have been successfully applied. The review also outlines future prospects for their use. Quantum impurity models describe an atom or molecule embedded in a host material, and are fundamental in nanoscience and correlated electron materials. These models are used to study quantum dots, molecular conductors, and the Kondo effect. The CT-QMC methods discussed here are particularly effective for a wide range of physically realistic models, providing access to both high and low energy scales. The article reviews several CT-QMC methods, including diagrammatic Monte Carlo, interaction expansion (CT-INT), auxiliary field (CT-AUX), hybridization expansion (CT-HYB), and infinite-U (CT-J) methods. Each method is described in detail, with their strengths and limitations discussed. The methods are particularly useful for solving problems involving many-body interactions, such as the Anderson impurity model and the Kondo lattice model. The review also discusses the challenges of simulating quantum impurity models, including the sign problem, which arises due to the complex nature of fermionic interactions. The CT-QMC methods are particularly effective in overcoming these challenges, allowing for accurate simulations of a wide range of physical systems. The article highlights the importance of CT-QMC methods in modern condensed matter physics, particularly in the study of strongly correlated electron systems. These methods have enabled the accurate simulation of complex systems, including those with multiple orbitals and interactions, and have been applied to a variety of problems in nanoscience and nonequilibrium physics. The review concludes with a discussion of the future prospects for CT-QMC methods, emphasizing their potential for further advancements in the study of quantum impurity models.